# Maximal/maximum independent sets

Is there something known about the class of graphs with the property that all maximal independent sets have the same cardinality and are therefore maximum ISs?

For example, take a set of points in the plane and consider the graph of intersections among all segments between pairs of points in the set. (segments->vertices, intersections->edges). This graph will have the above property, as all maximal ISs correspond to triangulations of the original point set. Are there other categories of graphs known to have this property? Can this property be easily tested?

-
There's a related paper here (portal.acm.org/citation.cfm?id=303085) that suggests that the problem of determining this for a given graph is co-NP-complete, and so characterizing the property will be tricky –  Suresh Venkat Aug 7 '11 at 2:32
Thank you for the reference! –  laszlo kozma Aug 8 '11 at 9:18

"It will be interesting to understand graphs where this property holds for all induced subgraphs". Unless I am mistaken, a (connected) graph and every of its induced subgraphs are well-covered iff it is a complete graph, since the path $P_3$ with 3 vertices and 2 edges is not well-covered. –  user13136 Feb 22 '13 at 18:55